Key takeaway: This precalculus trig graphs test covers amplitude, period, phase shift, midline, tangent graphs and sinusoidal models. It has nine questions for 20 points and should take about 45 minutes. Calculators are allowed, but give exact answers in terms of \(\pi\) unless told otherwise. Standard F-TF.B.5 guides the content, and two modeling questions use a dock and a Phoenix temperature.
This test covers amplitude, period, phase shift, midline, tangent graphs, and sinusoidal models. It has 9 questions worth 20 points and should take about 45 minutes. Calculators are allowed. Give exact answers in terms of \(\pi\) unless told otherwise.
- Grade: Precalculus
- Chapter: Graphs of Trigonometric Functions
- Common Core: F-TF.B.5, F-IF.C.7e, F-BF.B.3
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 2 pts; Question 3: 3 pts; Question 4: 2 pts; Question 5: 3 pts; Question 6: 3 pts; Question 7: 2 pts; Question 8: 2 pts; Question 9: 1 pt)
Precalculus Trig Graphs Test: questions 1–5
1. Features of a cosine (2 points)
- For \(y=-4\cos(\frac{x}{3})+2\), state the amplitude and the period. (1 point)
- State the midline, the maximum value, and the minimum value. (1 point)
2. Phase shift (2 points)
- For \(y=2\sin(3x-\pi)\), state the period and the phase shift. (1 point)
- Find the value of the function at \(x=\frac{\pi}{2}\). (1 point)
3. Equation from a graph (3 points)
The graph shows a cosine function.
- State the amplitude and the midline. (1 point)
- State the period. (1 point)
- Write an equation of the form \(y=A\cos(B(x-C))+D\). (1 point)
4. Sketch a sinusoid (2 points)
Sketch the graph of \(y=-2\sin 2x+1\) for \(0\le x\le\pi\) on the grid.
5. Tangent features (3 points)
Let \(y=3\tan 2x\).
- Find the period. (1 point)
- Find the two asymptotes closest to the y-axis. (1 point)
- Find the value at \(x=\frac{\pi}{8}\). (1 point)
Precalculus Trig Graphs Test: questions 6–9
6. Water depth model (3 points)
The graph shows the depth \(d\), in feet, of water at a dock. The depth is a sinusoid with high point \((2,16)\) and low point \((8,4)\).
- Find \(A\) and \(D\). (1 point)
- Write \(d(t)\) as a cosine function. (1 point)
- Find the depth at \(t=5\). (1 point)
7. Transformations (2 points)
Describe the transformations that take the graph of \(y=\sin x\) to the graph of \(y=-\frac{1}{2}\sin(x+\frac{\pi}{4})+3\).
8. Daily temperature (2 points)
In Phoenix the temperature peaks at \(104^{\circ}\text{F}\) at 3 p.m. (\(t=15\)) and is lowest at \(80^{\circ}\text{F}\) at 3 a.m. (\(t=3\)). Write a cosine model \(T(t)\) and find the temperature at 9 a.m. (2 points)
9. No amplitude (1 point)
Explain why the graph of \(y=\tan x\) has no amplitude. (1 point)
How do you pace the 20-point test?
Questions 3, 5 and 6 carry three points apiece, so give them the longest time. Meanwhile, questions 1, 2 and 4 are quick checks. Reserve the last five minutes to compare each graph with its equation.
What does question 2 test?
The function \(y=2\sin(3x-\pi)\) needs factoring first. Rewrite it as \(2\sin(3(x-\frac{\pi}{3}))\). As a result, the period is \(\frac{2\pi}{3}\) and the shift is \(\frac{\pi}{3}\) to the right.
Which precalculus trig graphs test question covers tangent?
Question 5 uses \(y=3\tan 2x\). The period is \(\frac{\pi}{2}\), and the asymptotes closest to the \(y\)-axis are at \(\pm\frac{\pi}{4}\). Question 9 then asks why tangent has no amplitude, so mention that it has no maximum or minimum value.
How do you build the temperature model in question 8?
The peak is 104 and the low is 80, so the amplitude is 12 and the midline is 92. Because the peak happens at \(t=15\), cosine with a shift of 15 fits well. Also, the period is 24 hours.
What should you do after you finish the graph test?
Grade yourself with the key and mark every lost point as a feature error or a shift error. Next, redraw the sketch from question 4 without notes. Finally, write one new equation from a graph, then check it with a calculator.
See every chapter on the Precalculus math page.
